• Previous Article
    Dissipative quasi-geostrophic equations in critical Sobolev spaces: Smoothing effect and global well-posedness
  • DCDS Home
  • This Issue
  • Next Article
    Singular perturbation systems with stochastic forcing and the renormalization group method
2010, 26(4): 1213-1240. doi: 10.3934/dcds.2010.26.1213

Boundary layers in smooth curvilinear domains: Parabolic problems

1. 

The Institute for Scientific Computing and Applied Mathematics, Indiana University, 831 East Third Street, Bloomington, Indiana 47405, United States, United States

2. 

Institute for Scientific Computing and Applied Mathematics, Indiana University, Rawles Hall, 831 E. Third Street, Bloomington, IN 47405

Received  March 2009 Revised  May 2009 Published  December 2009

The goal of this article is to study the boundary layer of the heat equation with thermal diffusivity in a general (curved), bounded and smooth domain in $\mathbb{R}^{d}$, $d \geq 2$, when the diffusivity parameter ε is small. Using a curvilinear coordinate system fitting the boundary, an asymptotic expansion, with respect to ε, of the heat solution is obtained at all orders. It appears that unlike the case of a straight boundary, because of the curvature of the boundary, two correctors in powers of ε and ε1/2 must be introduced at each order. The convergence results, between the exact and approximate solutions, seem optimal. Beside the intrinsic interest of the results presented in the article, we believe that some of the methods introduced here should be useful to study boundary layers for other problems involving curved boundaries.
Citation: Gung-Min Gie, Makram Hamouda, Roger Témam. Boundary layers in smooth curvilinear domains: Parabolic problems. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1213-1240. doi: 10.3934/dcds.2010.26.1213
[1]

François Monard. Efficient tensor tomography in fan-beam coordinates. Ⅱ: Attenuated transforms. Inverse Problems & Imaging, 2018, 12 (2) : 433-460. doi: 10.3934/ipi.2018019

[2]

Bernard Brighi, Tewfik Sari. Blowing-up coordinates for a similarity boundary layer equation. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 929-948. doi: 10.3934/dcds.2005.12.929

[3]

Hongyun Peng, Zhi-An Wang, Kun Zhao, Changjiang Zhu. Boundary layers and stabilization of the singular Keller-Segel system. Kinetic & Related Models, 2018, 11 (5) : 1085-1123. doi: 10.3934/krm.2018042

[4]

Umberto Biccari. Boundary controllability for a one-dimensional heat equation with a singular inverse-square potential. Mathematical Control & Related Fields, 2018, 8 (0) : 1-29. doi: 10.3934/mcrf.2019011

[5]

Dominika Pilarczyk. Asymptotic stability of singular solution to nonlinear heat equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 991-1001. doi: 10.3934/dcds.2009.25.991

[6]

Niclas Bernhoff. Boundary layers and shock profiles for the discrete Boltzmann equation for mixtures. Kinetic & Related Models, 2012, 5 (1) : 1-19. doi: 10.3934/krm.2012.5.1

[7]

Abdelhakim Belghazi, Ferroudja Smadhi, Nawel Zaidi, Ouahiba Zair. Carleman inequalities for the two-dimensional heat equation in singular domains. Mathematical Control & Related Fields, 2012, 2 (4) : 331-359. doi: 10.3934/mcrf.2012.2.331

[8]

Jing Wang, Lining Tong. Vanishing viscosity limit of 1d quasilinear parabolic equation with multiple boundary layers. Communications on Pure & Applied Analysis, 2019, 18 (2) : 887-910. doi: 10.3934/cpaa.2019043

[9]

R. Estrada. Boundary layers and spectral content asymptotics. Conference Publications, 1998, 1998 (Special) : 242-252. doi: 10.3934/proc.1998.1998.242

[10]

Chang-Yeol Jung, Roger Temam. Interaction of boundary layers and corner singularities. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 315-339. doi: 10.3934/dcds.2009.23.315

[11]

Yu Gao, Jian-Guo Liu. The modified Camassa-Holm equation in Lagrangian coordinates. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2545-2592. doi: 10.3934/dcdsb.2018067

[12]

Keng Deng, Zhihua Dong. Blow-up for the heat equation with a general memory boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2147-2156. doi: 10.3934/cpaa.2012.11.2147

[13]

Carmen Calvo-Jurado, Juan Casado-Díaz, Manuel Luna-Laynez. The homogenization of the heat equation with mixed conditions on randomly subsets of the boundary. Conference Publications, 2013, 2013 (special) : 85-94. doi: 10.3934/proc.2013.2013.85

[14]

Huicong Li. Effective boundary conditions of the heat equation on a body coated by functionally graded material. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1415-1430. doi: 10.3934/dcds.2016.36.1415

[15]

Kazuhiro Ishige, Ryuichi Sato. Heat equation with a nonlinear boundary condition and uniformly local $L^r$ spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2627-2652. doi: 10.3934/dcds.2016.36.2627

[16]

Zvi Artstein. Invariance principle in the singular perturbations limit. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-14. doi: 10.3934/dcdsb.2018309

[17]

Alain Miranville, Sergey Zelik. The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 275-310. doi: 10.3934/dcds.2010.28.275

[18]

Aníbal Rodríguez-Bernal, Enrique Zuazua. Parabolic singular limit of a wave equation with localized boundary damping. Discrete & Continuous Dynamical Systems - A, 1995, 1 (3) : 303-346. doi: 10.3934/dcds.1995.1.303

[19]

P. Lima, L. Morgado. Analysis of singular boundary value problems for an Emden-Fowler equation. Communications on Pure & Applied Analysis, 2006, 5 (2) : 321-336. doi: 10.3934/cpaa.2006.5.321

[20]

Zongming Guo, Yunting Yu. Boundary value problems for a semilinear elliptic equation with singular nonlinearity. Communications on Pure & Applied Analysis, 2016, 15 (2) : 399-412. doi: 10.3934/cpaa.2016.15.399

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]